login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A141307 Number of bicolored connected permutations. 3

%I

%S 2,4,24,208,2272,29504,441216,7447808,139951616,2897228800,

%T 65533753344,1608679247872,42607095439360,1211489065582592,

%U 36818002833014784,1191230067009978368,40888060455008731136,1484180363633916903424,56809679459301490950144,2287045885619374501396480,96608773951155028111654912

%N Number of bicolored connected permutations.

%C Number of generators of degree n of the Hopf algebra of free quasi-symmetric functions of level 2. For level r, this would be r^n*c(n), where c(n) is the number of connected permutations (A003319).

%C These are also the dimensions of the graded components of the primitive Lie algebra of the same Hopf algebra.

%H J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008.

%F a(n) = 2^n * A003319(n).

%F G.f.: 1/x - Q(0)/x where Q(k) = 1 - 2*x*(k+1)/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Apr 02 2013

%F G.f.: 1/x - (1 + x)/x/(x*Q(0) + 1) where Q(k)= 1 + (2*k+2)/(1 - x/(x + 1/Q(k+1) )); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 11 2013

%F G.f.: 1/x - G(0)/(2*x), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(2*k+2)) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 29 2013

%e a(1)=2 because there are two colorings of the permutation (1).

%p 2^n*op(n,INVERTi([seq(k!, k=1..n)]))

%t a3319[0] = 0; a3319[n_] := a3319[n] = n! - Sum[k! a3319[n-k], {k, 1, n-1}];

%t a[n_] := 2^n a3319[n];

%t Array[a, 21] (* _Jean-François Alcover_, Dec 10 2018 *)

%Y Cf. A003319.

%K nonn

%O 1,1

%A Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 16:17 EST 2021. Contains 349445 sequences. (Running on oeis4.)